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source: Centre International de Rencontres Mathématiques 2016年11月2日

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 1 1:27:36

Abstract: * The early results of Ramsey theory:

Hilbert's irreducibility theorem, Dickson-Schur work on Fermat's equation over finite fields, van der Waerden's theorem, Ramsey's theorem and its rediscovery by Erdos and Szekeres.

* Three main principles of Ramsey theory:

First principle: Complete disorder is impossible. Second principle: Behind every 'Partition' result there is a notion of largeness which is responsible for a 'Density' enhancement of this result. Third principle: The sought-after configurations which are always to be found in large sets are abundant.

* Furstenberg's Dynamical approach:

Partition Ramsey theory and topological dynamics Dynamical versions of van der Waerden's theorem, Hindman's theorem and Graham-Rothschild-Spencer's geometric Ramsey. Density Ramsey theory and Furstenberg's correspondence principle Furstenberg's correspondence principle. Ergodic Szemeredi's theorem. Polynomial Szemeredi theorem. Density version of the Hales-Jewett theorem.

* Stone-Cech compactifications and Hindman's theorem:

Topological algebra in Stone-Cech compactifications. Proof of Hind-man's theorem via Poincare recurrence theorem for ultrafilters.

* IP sets and ergodic Ramsey theory:

Applications of IP sets and idempotent ultrafilters to ergodic-theoretical multiple recurrence and to density Ramsey theory. IP-polynomial Szemeredi theorem.

* Open problems and conjectures

If time permits: * The nilpotent connection, * Ergodic Ramsey theory and amenable groups

Recording during the Jean-Morlet Chair doctoral school: "Spectral Theory of Dynamical Systems and Related Topics" the October 17, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 2 1:30:01

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 3 1:12:02

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics- part 4 1:01:01

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 5 1:26:29

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 6 1:30:33

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 7 1:13:51

Vitaly Bergelson: Mutually enriching connections between ergodic theory and combinatorics - part 8 58:42

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